A quality characteristic is monitored by a control chart designed so that the probability that a certain out-of-control condition will be detected on the first sample following the shift to that state is 1-B . Find the following:

(i) The probability that the out-of-control condition will be detected on the second sample following the shift.

(ii) The probability that the out-of-control condition will be detected on the k th sample following the shift.

(iii) The expected number of subgroups analyzed before the shift is detected. (iv) The probability that the first sample following the shift produces a statistic that plots inside the control limits

. (v) The probability that the shift is detected on the k th subsequent sample. (vi) The probability that two consecutive samples produce statistics that plot outside of the control limits.

(vii) The probability that at least one of the next k samples yields values of the test statistic that plot outside the control limits.